Additionally to [1]}, we show how one-dimensional discretized problems with a velocity that can change its sign several times in domain can be solved by one forward and one backward substitution as known for so called fast sweeping methods [2]}, [3]}. The derivation of second order scheme is based here on the discretization of an error term for the fully implicit first order accurate upwind scheme. Consequently, it is easy to apply some limiter procedures based on a blending between these two types of schemes if necessary. The free parameter of this family of schemes is introduced in a more convenient way than in [1]}. The parameter can vary in space and time that we demonstrate in one numerical experiment by letting find its optimized values using differential programming tools.
